Subgroups — C₇×#24772 #13201988

Order 1792.

840 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#473	C₂	1	2	yes	maximal
2	C₇×#3039	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#462	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₇×#137	C₂	1	1	yes	maximal, characteristic
5	C₇×#2906	C₂	1	1	yes	maximal, characteristic
6	C₇×#2064	C₂	1	2	yes	maximal, complemented
7	C₇×#2047	C₂	1	2	yes	maximal, complemented
8	C₇×#1966	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1950	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1908	C₂	1	2	yes	maximal
11	C₇×#1801	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#137	C₂²	1	2	yes	—
13	C₂×C₇×#137	C₂²	1	1	yes	characteristic
14	C₂²×C₇×Q₁₆	C₂²	1	2	yes	—
15	C₂×C₇×#131	C₂²	1	2	yes	—
16	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₄×C₇×#41	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
19	C₇×#473	—	2	8	no	—
20	C₇×#462	—	2	1	no	—
21	C₇×#462	—	2	1	no	—
22	C₇×#462	—	2	1	no	—
23	C₇×#462	—	2	1	no	—
24	C₇×#462	—	2	2	no	—
25	C₇×#461	—	2	1	no	—
26	C₇×#461	—	2	1	no	—
27	C₇×#441	—	2	1	no	—
28	C₇×#440	—	2	1	no	—
29	C₄×C₇×Q₁₆	—	2	1	no	—
30	C₄×C₇×SD₁₆	—	2	1	no	—
31	C₂×C₇×#108	C₂²	1	2	yes	—
32	C₇×#428	C₂²	1	1	yes	characteristic
33	C₇×#424	—	2	2	no	—
34	C₇×#424	C₂²	1	2	yes	—
35	C₇×#423	—	2	1	no	—
36	C₇×#423	C₂²	1	1	yes	characteristic
37	C₇×#423	C₂²	1	1	yes	characteristic
38	C₂×C₇×C₄⋊C₈	—	2	1	no	—
39	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₇×#419	C₂²	1	1	yes	characteristic
41	C₇×#419	C₂²	1	2	yes	—
42	C₇×#417	C₂²	1	1	yes	characteristic
43	C₂×C₇×#103	C₂²	1	2	yes	—
44	C₂×C₇×#101	C₂²	1	2	yes	—
45	C₂×C₇×#101	C₂²	1	2	yes	—
46	C₂×C₇×#101	C₂²	1	2	yes	—
47	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
48	C₇×#343	C₂²	1	2	yes	—
49	C₇×#341	C₂²	1	2	yes	—
50	#24772	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂×C₇×#41	C₂³	1	1	yes	characteristic
52	C₂²×C₇×Q₈	C₂³	1	2	yes	—
53	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
54	C₇×#137	—	2	2	no	—
55	C₇×#137	—	2	4	no	—
56	C₇×#137	—	2	4	no	—
57	C₇×#137	D₄	1	4	yes	—
58	C₂×C₇×Q₁₆	—	2	1	no	—
59	C₂×C₇×Q₁₆	—	2	4	no	—
60	C₂×C₇×Q₁₆	—	2	2	no	—
61	C₂×C₇×Q₁₆	—	2	2	no	—
62	C₂×C₇×Q₁₆	—	2	2	no	—
63	C₂×C₇×Q₁₆	—	4	2	no	—
64	C₂×C₇×SD₁₆	—	2	1	no	—
65	C₂×C₇×SD₁₆	—	2	2	no	—
66	C₇×#131	—	2	4	no	—
67	C₇×#131	—	4	2	no	—
68	C₂×C₇×M₁₆	—	2	2	no	—
69	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
70	C₂×C₇×M₁₆	C₂³	1	2	yes	—
71	C₂×C₇×M₁₆	C₂³	1	2	yes	—
72	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
73	C₂²×C₇×C₈	—	2	2	no	—
74	C₂²×C₇×C₈	C₂³	1	2	yes	—
75	C₇×C₄⋊Q₈	—	2	1	no	—
76	C₇×C₄⋊Q₈	—	2	1	no	—
77	C₇×C₄⋊Q₈	—	2	1	no	—
78	C₇×C₄⋊Q₈	—	2	1	no	—
79	C₄×C₇×Q₈	—	2	1	no	—
80	C₄×C₇×Q₈	—	2	1	no	—
81	C₄×C₇×Q₈	—	2	1	no	—
82	C₄×C₇×Q₈	—	2	1	no	—
83	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
84	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
85	C₄×C₇×D₄	—	2	1	no	—
86	C₄×C₇×D₄	—	2	1	no	—
87	C₄×C₇×D₄	D₄	1	1	yes	characteristic
88	C₄×C₇×D₄	D₄	1	1	yes	characteristic
89	C₇×#117	—	2	1	no	—
90	C₇×#117	C₂³	1	1	yes	characteristic
91	C₂×C₇×C₄⋊C₄	—	2	1	no	—
92	C₂×C₇×C₄⋊C₄	—	2	1	no	—
93	C₂×C₇×C₄⋊C₄	—	2	1	no	—
94	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₇	—	2	1	no	—
96	C₂×C₄²×C₇	—	2	1	no	—
97	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
98	C₇×#108	—	2	4	no	—
99	C₇×#107	—	2	1	no	—
100	C₇×#106	—	2	1	no	—
101	C₇×C₄⋊C₈	—	2	2	no	—
102	C₇×C₄⋊C₈	—	2	2	no	—
103	C₇×C₄⋊C₈	—	2	1	no	—
104	C₇×C₄⋊C₈	—	2	1	no	—
105	C₇×C₄⋊C₈	—	2	1	no	—
106	C₇×C₄⋊C₈	—	2	1	no	—
107	C₇×C₄⋊C₈	D₄	1	1	yes	characteristic
108	C₇×C₄⋊C₈	D₄	1	1	yes	characteristic
109	C₇×C₄⋊C₈	D₄	1	2	yes	—
110	C₇×#103	—	2	1	no	—
111	C₇×#103	—	2	2	no	—
112	C₇×#103	—	2	2	no	—
113	C₇×#103	—	2	1	no	—
114	C₇×#103	—	2	1	no	—
115	C₇×#103	—	2	2	no	—
116	C₇×#103	—	2	2	no	—
117	C₇×#102	—	2	1	no	—
118	C₇×#101	—	2	4	no	—
119	C₇×#101	—	2	4	no	—
120	C₇×#101	—	4	2	no	—
121	C₇×#97	—	2	1	no	—
122	C₄×C₇×C₈	—	2	1	no	—
123	C₂×#473	C₂×C₇	1	2	yes	—
124	#3039	C₂×C₇	1	1	yes	characteristic
125	C₂×#462	C₂×C₇	1	1	yes	characteristic, complemented
126	C₄×#137	C₂×C₇	1	1	yes	characteristic
127	#2906	C₂×C₇	1	1	yes	characteristic
128	#2064	C₂×C₇	1	2	yes	complemented
129	#2047	C₂×C₇	1	2	yes	complemented
130	#1966	C₂×C₇	1	1	yes	characteristic, complemented
131	#1950	C₂×C₇	1	1	yes	characteristic, complemented
132	#1908	C₂×C₇	1	2	yes	—
133	#1801	C₂×C₇	1	1	yes	characteristic, complemented
134	C₇×#41	—	2	1	no	—
135	C₇×#41	—	2	1	no	—
136	C₇×#41	C₂×D₄	1	2	yes	—
137	C₇×#41	#41	1	2	yes	—
138	C₂×C₇×Q₈	—	2	1	no	—
139	C₂×C₇×Q₈	—	2	1	no	—
140	C₂×C₇×Q₈	—	2	1	no	—
141	C₂×C₇×Q₈	—	2	2	no	—
142	C₂×C₇×Q₈	—	2	2	no	—
143	C₂×C₇×Q₈	—	2	2	no	—
144	C₂×C₇×Q₈	—	2	1	no	—
145	C₂×C₇×Q₈	—	2	2	no	—
146	C₂×C₇×Q₈	—	2	1	no	—
147	C₂×C₇×Q₈	—	2	2	no	—
148	C₂×C₇×Q₈	—	2	2	no	—
149	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
150	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
151	C₂×C₇×D₄	—	2	1	no	—
152	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₇	—	2	1	no	—
154	C₂²×C₄×C₇	—	2	1	no	—
155	C₂²×C₄×C₇	—	2	1	no	—
156	C₂²×C₄×C₇	—	2	1	no	—
157	C₂²×C₄×C₇	—	2	2	no	—
158	C₂²×C₄×C₇	—	2	1	no	—
159	C₂²×C₄×C₇	#41	1	1	yes	characteristic
160	C₂²×C₄×C₇	#41	1	1	yes	characteristic
161	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
162	C₇×Q₁₆	—	2	4	no	—
163	C₇×Q₁₆	—	4	4	no	—
164	C₇×Q₁₆	—	4	4	no	—
165	C₇×Q₁₆	—	4	4	no	—
166	C₇×SD₁₆	—	2	4	no	—
167	C₇×SD₁₆	—	4	4	no	—
168	C₇×M₁₆	—	2	1	no	—
169	C₇×M₁₆	—	2	2	no	—
170	C₇×M₁₆	—	2	2	no	—
171	C₇×M₁₆	—	2	2	no	—
172	C₇×M₁₆	—	2	1	no	—
173	C₇×M₁₆	—	2	2	no	—
174	C₇×M₁₆	—	2	1	no	—
175	C₇×M₁₆	—	4	4	no	—
176	C₇×M₁₆	C₂×D₄	1	2	yes	—
177	C₂×C₇×C₈	—	2	1	no	—
178	C₂×C₇×C₈	—	2	2	no	—
179	C₂×C₇×C₈	—	2	2	no	—
180	C₂×C₇×C₈	—	2	1	no	—
181	C₂×C₇×C₈	—	2	2	no	—
182	C₂×C₇×C₈	—	2	2	no	—
183	C₂×C₇×C₈	—	4	4	no	—
184	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
185	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
186	C₇×C₄⋊C₄	—	2	1	no	—
187	C₇×C₄⋊C₄	—	2	1	no	—
188	C₇×C₄⋊C₄	—	2	1	no	—
189	C₇×C₄⋊C₄	—	2	1	no	—
190	C₇×C₄⋊C₄	—	4	1	no	—
191	C₇×C₄⋊C₄	—	4	1	no	—
192	C₇×C₄⋊C₄	—	4	1	no	—
193	C₇×C₄⋊C₄	—	4	1	no	—
194	C₇×C₄⋊C₄	—	4	1	no	—
195	C₇×C₄⋊C₄	—	4	1	no	—
196	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
197	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
198	C₇×C₂²⋊C₄	—	2	1	no	—
199	C₇×C₂²⋊C₄	—	2	1	no	—
200	C₇×C₂²⋊C₄	—	2	1	no	—
201	C₄²×C₇	—	2	1	no	—
202	C₄²×C₇	—	2	1	no	—
203	C₄²×C₇	—	2	1	no	—
204	C₄²×C₇	—	4	1	no	—
205	C₄²×C₇	—	4	1	no	—
206	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
207	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
208	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
209	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
210	C₂×#137	C₂²×C₇	1	2	yes	—
211	C₂×#137	C₂²×C₇	1	1	yes	characteristic
212	C₂²×Q₁₆	C₂²×C₇	1	2	yes	—
213	C₂×#131	C₂²×C₇	1	2	yes	—
214	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
215	C₄×#41	C₂²×C₇	1	1	yes	characteristic
216	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
217	#473	—	2	8	no	—
218	#462	—	2	1	no	—
219	#462	—	2	1	no	—
220	#462	—	2	1	no	—
221	#462	—	2	1	no	—
222	#462	—	2	2	no	—
223	#461	—	2	1	no	—
224	#461	—	2	1	no	—
225	#441	—	2	1	no	—
226	#440	—	2	1	no	—
227	C₄×Q₁₆	—	2	1	no	—
228	C₄×SD₁₆	—	2	1	no	—
229	C₂×#108	C₂²×C₇	1	2	yes	—
230	#428	C₂²×C₇	1	1	yes	characteristic
231	#424	—	2	2	no	—
232	#424	C₂²×C₇	1	2	yes	—
233	#423	—	2	1	no	—
234	#423	C₂²×C₇	1	1	yes	characteristic
235	#423	C₂²×C₇	1	1	yes	characteristic
236	C₂×C₄⋊C₈	—	2	1	no	—
237	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
238	#419	C₂²×C₇	1	2	yes	—
239	#419	C₂²×C₇	1	1	yes	characteristic
240	#417	C₂²×C₇	1	1	yes	characteristic
241	C₂×#103	C₂²×C₇	1	2	yes	—
242	C₂×#101	C₂²×C₇	1	2	yes	—
243	C₂×#101	C₂²×C₇	1	2	yes	—
244	C₂×#101	C₂²×C₇	1	2	yes	—
245	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
246	#343	C₂²×C₇	1	2	yes	—
247	#341	C₂²×C₇	1	2	yes	—
248	C₂³×C₇	—	2	1	no	—
249	C₂³×C₇	C₂×#41	1	1	yes	characteristic
250	C₇×Q₈	—	2	1	no	—
251	C₇×Q₈	—	2	2	no	—
252	C₇×Q₈	—	2	2	no	—
253	C₇×Q₈	—	4	1	no	—
254	C₇×Q₈	—	4	1	no	—
255	C₇×Q₈	—	4	2	no	—
256	C₇×Q₈	—	4	2	no	—
257	C₇×Q₈	—	4	2	no	—
258	C₇×Q₈	—	4	2	no	—
259	C₇×Q₈	#121	1	2	yes	—
260	C₇×D₄	—	2	2	no	—
261	C₇×D₄	—	2	2	no	—
262	C₇×D₄	—	2	1	no	—
263	C₇×D₄	#121	1	2	yes	—
264	C₂×C₄×C₇	—	2	1	no	—
265	C₂×C₄×C₇	—	2	2	no	—
266	C₂×C₄×C₇	—	2	2	no	—
267	C₂×C₄×C₇	—	2	2	no	—
268	C₂×C₄×C₇	—	2	1	no	—
269	C₂×C₄×C₇	—	2	1	no	—
270	C₂×C₄×C₇	—	2	2	no	—
271	C₂×C₄×C₇	—	4	1	no	—
272	C₂×C₄×C₇	—	4	1	no	—
273	C₂×C₄×C₇	—	4	1	no	—
274	C₂×C₄×C₇	—	4	2	no	—
275	C₂×C₄×C₇	—	4	1	no	—
276	C₂×C₄×C₇	—	4	2	no	—
277	C₂×C₄×C₇	—	4	2	no	—
278	C₂×C₄×C₇	—	4	2	no	—
279	C₂×C₄×C₇	—	8	1	no	—
280	C₂×C₄×C₇	—	8	1	no	—
281	C₂×C₄×C₇	#121	1	1	yes	characteristic
282	C₂×C₄×C₇	#121	1	1	yes	characteristic
283	C₂×C₄×C₇	#121	1	1	yes	characteristic
284	C₂×C₄×C₇	#121	1	1	yes	characteristic
285	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
286	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
287	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
288	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
289	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
290	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
291	C₇×C₈	—	2	2	no	—
292	C₇×C₈	—	2	2	no	—
293	C₇×C₈	—	2	2	no	—
294	C₇×C₈	—	4	2	no	—
295	C₇×C₈	—	4	1	no	—
296	C₇×C₈	—	4	2	no	—
297	C₂×#41	C₂³×C₇	1	1	yes	characteristic
298	C₂²×Q₈	C₂³×C₇	1	2	yes	—
299	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
300	#137	—	2	2	no	—
301	#137	—	2	4	no	—
302	#137	—	2	4	no	—
303	#137	C₇×D₄	1	4	yes	—
304	C₂×Q₁₆	—	2	1	no	—
305	C₂×Q₁₆	—	2	4	no	—
306	C₂×Q₁₆	—	2	2	no	—
307	C₂×Q₁₆	—	2	2	no	—
308	C₂×Q₁₆	—	2	2	no	—
309	C₂×Q₁₆	—	4	2	no	—
310	C₂×SD₁₆	—	2	1	no	—
311	C₂×SD₁₆	—	2	2	no	—
312	#131	—	2	4	no	—
313	#131	—	4	2	no	—
314	C₂×M₁₆	—	2	2	no	—
315	C₂×M₁₆	C₂³×C₇	1	2	yes	—
316	C₂×M₁₆	C₂³×C₇	1	2	yes	—
317	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
318	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
319	C₂²×C₈	—	2	2	no	—
320	C₂²×C₈	C₂³×C₇	1	2	yes	—
321	C₄⋊Q₈	—	2	1	no	—
322	C₄⋊Q₈	—	2	1	no	—
323	C₄⋊Q₈	—	2	1	no	—
324	C₄⋊Q₈	—	2	1	no	—
325	C₄×Q₈	—	2	1	no	—
326	C₄×Q₈	—	2	1	no	—
327	C₄×Q₈	—	2	1	no	—
328	C₄×Q₈	—	2	1	no	—
329	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
330	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
331	C₄×D₄	—	2	1	no	—
332	C₄×D₄	—	2	1	no	—
333	C₄×D₄	C₇×D₄	1	1	yes	characteristic
334	C₄×D₄	C₇×D₄	1	1	yes	characteristic
335	#117	—	2	1	no	—
336	#117	C₂³×C₇	1	1	yes	characteristic
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	—	2	1	no	—
340	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
341	C₂×C₄²	—	2	1	no	—
342	C₂×C₄²	—	2	1	no	—
343	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
344	#108	—	2	4	no	—
345	#107	—	2	1	no	—
346	#106	—	2	1	no	—
347	C₄⋊C₈	—	2	2	no	—
348	C₄⋊C₈	—	2	2	no	—
349	C₄⋊C₈	—	2	1	no	—
350	C₄⋊C₈	—	2	1	no	—
351	C₄⋊C₈	—	2	1	no	—
352	C₄⋊C₈	—	2	1	no	—
353	C₄⋊C₈	C₇×D₄	1	1	yes	characteristic
354	C₄⋊C₈	C₇×D₄	1	1	yes	characteristic
355	C₄⋊C₈	C₇×D₄	1	2	yes	—
356	#103	—	2	1	no	—
357	#103	—	2	2	no	—
358	#103	—	2	2	no	—
359	#103	—	2	1	no	—
360	#103	—	2	1	no	—
361	#103	—	2	2	no	—
362	#103	—	2	2	no	—
363	#102	—	2	1	no	—
364	#101	—	2	4	no	—
365	#101	—	2	4	no	—
366	#101	—	4	2	no	—
367	#97	—	2	1	no	—
368	C₄×C₈	—	2	1	no	—
369	C₂²×C₇	—	2	1	no	—
370	C₂²×C₇	—	2	1	no	—
371	C₂²×C₇	—	2	2	no	—
372	C₂²×C₇	—	4	1	no	—
373	C₂²×C₇	—	4	1	no	—
374	C₂²×C₇	C₂×#121	1	1	yes	characteristic
375	C₂²×C₇	#547	1	1	yes	characteristic
376	C₂²×C₇	#553	1	1	yes	characteristic
377	C₄×C₇	—	2	2	no	—
378	C₄×C₇	—	2	1	no	—
379	C₄×C₇	—	2	1	no	—
380	C₄×C₇	—	2	1	no	—
381	C₄×C₇	—	4	2	no	—
382	C₄×C₇	—	4	1	no	—
383	C₄×C₇	—	4	1	no	—
384	C₄×C₇	—	8	2	no	—
385	C₄×C₇	—	8	1	no	—
386	C₄×C₇	#473	1	2	yes	complemented
387	C₄×C₇	C₂×#121	1	1	yes	characteristic
388	C₄×C₇	#537	1	1	yes	characteristic
389	C₄×C₇	D₄²	1	1	yes	characteristic
390	C₄×C₇	#548	1	1	yes	characteristic
391	#41	—	2	1	no	—
392	#41	—	2	1	no	—
393	#41	C₂×C₇×D₄	1	2	yes	—
394	#41	C₇×#41	1	2	yes	—
395	C₂×Q₈	—	2	1	no	—
396	C₂×Q₈	—	2	1	no	—
397	C₂×Q₈	—	2	1	no	—
398	C₂×Q₈	—	2	2	no	—
399	C₂×Q₈	—	2	2	no	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	2	1	no	—
402	C₂×Q₈	—	2	2	no	—
403	C₂×Q₈	—	2	1	no	—
404	C₂×Q₈	—	2	2	no	—
405	C₂×Q₈	—	2	2	no	—
406	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
407	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
408	C₂×D₄	—	2	1	no	—
409	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
410	C₂²×C₄	—	2	1	no	—
411	C₂²×C₄	—	2	1	no	—
412	C₂²×C₄	—	2	1	no	—
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	C₇×#41	1	1	yes	characteristic
417	C₂²×C₄	C₇×#41	1	1	yes	characteristic
418	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
419	Q₁₆	—	2	4	no	—
420	Q₁₆	—	4	4	no	—
421	Q₁₆	—	4	4	no	—
422	Q₁₆	—	4	4	no	—
423	SD₁₆	—	2	4	no	—
424	SD₁₆	—	4	4	no	—
425	M₁₆	—	2	1	no	—
426	M₁₆	—	2	2	no	—
427	M₁₆	—	2	2	no	—
428	M₁₆	—	2	2	no	—
429	M₁₆	—	2	1	no	—
430	M₁₆	—	2	2	no	—
431	M₁₆	—	2	1	no	—
432	M₁₆	—	4	4	no	—
433	M₁₆	C₂×C₇×D₄	1	2	yes	—
434	C₂×C₈	—	2	1	no	—
435	C₂×C₈	—	2	2	no	—
436	C₂×C₈	—	2	2	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	2	2	no	—
439	C₂×C₈	—	2	2	no	—
440	C₂×C₈	—	4	4	no	—
441	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
442	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
443	C₄⋊C₄	—	2	1	no	—
444	C₄⋊C₄	—	2	1	no	—
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	4	1	no	—
448	C₄⋊C₄	—	4	1	no	—
449	C₄⋊C₄	—	4	1	no	—
450	C₄⋊C₄	—	4	1	no	—
451	C₄⋊C₄	—	4	1	no	—
452	C₄⋊C₄	—	4	1	no	—
453	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
454	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
455	C₂²⋊C₄	—	2	1	no	—
456	C₂²⋊C₄	—	2	1	no	—
457	C₂²⋊C₄	—	2	1	no	—
458	C₄²	—	2	1	no	—
459	C₄²	—	2	1	no	—
460	C₄²	—	2	1	no	—
461	C₄²	—	4	1	no	—
462	C₄²	—	4	1	no	—
463	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
464	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
465	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
466	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
467	C₂×C₇	—	2	1	no	—
468	C₂×C₇	—	2	1	no	—
469	C₂×C₇	—	4	2	no	—
470	C₂×C₇	#2416	1	1	yes	characteristic
471	C₂×C₇	C₂×#473	1	1	yes	characteristic
472	C₂×C₇	#3069	1	1	yes	characteristic
473	C₂³	—	2	1	no	—
474	C₂³	C₂×C₇×#41	1	1	yes	characteristic
475	Q₈	—	2	1	no	—
476	Q₈	—	2	2	no	—
477	Q₈	—	2	2	no	—
478	Q₈	—	4	1	no	—
479	Q₈	—	4	1	no	—
480	Q₈	—	4	2	no	—
481	Q₈	—	4	2	no	—
482	Q₈	—	4	2	no	—
483	Q₈	—	4	2	no	—
484	Q₈	C₇×#121	1	2	yes	—
485	D₄	—	2	2	no	—
486	D₄	—	2	2	no	—
487	D₄	—	2	1	no	—
488	D₄	C₇×#121	1	2	yes	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	4	1	no	—
497	C₂×C₄	—	4	1	no	—
498	C₂×C₄	—	4	1	no	—
499	C₂×C₄	—	4	2	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	—	4	2	no	—
504	C₂×C₄	—	8	1	no	—
505	C₂×C₄	—	8	1	no	—
506	C₂×C₄	C₇×#121	1	1	yes	characteristic
507	C₂×C₄	C₇×#121	1	1	yes	characteristic
508	C₂×C₄	C₇×#121	1	1	yes	characteristic
509	C₂×C₄	C₇×#121	1	1	yes	characteristic
510	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
511	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
512	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
513	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
514	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
515	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
516	C₈	—	2	2	no	—
517	C₈	—	2	2	no	—
518	C₈	—	2	2	no	—
519	C₈	—	4	2	no	—
520	C₈	—	4	1	no	—
521	C₈	—	4	2	no	—
522	C₇	#24772	1	1	yes	characteristic, direct product factor, Sylow
523	C₂²	—	2	1	no	—
524	C₂²	—	2	1	no	—
525	C₂²	—	2	2	no	—
526	C₂²	—	4	1	no	—
527	C₂²	—	4	1	no	—
528	C₂²	C₂×C₇×#121	1	1	yes	characteristic
529	C₂²	C₇×#547	1	1	yes	characteristic
530	C₂²	C₇×#553	1	1	yes	characteristic
531	C₄	—	2	2	no	—
532	C₄	—	2	1	no	—
533	C₄	—	2	1	no	—
534	C₄	—	2	1	no	—
535	C₄	—	4	2	no	—
536	C₄	—	4	1	no	—
537	C₄	—	4	1	no	—
538	C₄	—	8	2	no	—
539	C₄	—	8	1	no	—
540	C₄	C₇×#473	1	2	yes	complemented
541	C₄	C₂×C₇×#121	1	1	yes	characteristic
542	C₄	C₇×#537	1	1	yes	characteristic
543	C₄	C₇×D₄²	1	1	yes	characteristic
544	C₄	C₇×#548	1	1	yes	characteristic
545	C₂	—	2	1	no	—
546	C₂	—	2	1	no	—
547	C₂	—	4	2	no	—
548	C₂	C₇×#2416	1	1	yes	characteristic
549	C₂	C₂×C₇×#473	1	1	yes	characteristic
550	C₂	C₇×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂×C₇×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₂×C₇×#462	C₂	semidirect
6. C₇×#2064	C₂	semidirect
7. C₇×#2047	C₂	semidirect
8. C₇×#1966	C₂	semidirect
9. C₇×#1950	C₂	semidirect
11. C₇×#1801	C₂	semidirect
50. #24772	C₇	direct (N × H)
125. C₂×#462	C₂×C₇	semidirect
128. #2064	C₂×C₇	semidirect
129. #2047	C₂×C₇	semidirect
130. #1966	C₂×C₇	semidirect
131. #1950	C₂×C₇	semidirect
133. #1801	C₂×C₇	semidirect
386. C₄×C₇	#473	semidirect
522. C₇	#24772	direct (N × H)
540. C₄	C₇×#473	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×#473	2
C₇×#3039	2
C₂×C₇×#462	2
C₄×C₇×#137	2
C₇×#2906	2
C₇×#2064	2
C₇×#2047	2
C₇×#1966	2
C₇×#1950	2
C₇×#1908	2
C₇×#1801	2
#24772	7

Subgroups — C7×#24772 #13201988

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#24772 #13201988